Let's go through the problem based on the image you uploaded.

The setup is a hypothesis test for the proportion of voters who prefer Candidate A, with the following hypotheses:

• Null Hypothesis ($H_0$): $p = 0.54$
• Alternative Hypothesis ($H_1$): $p < 0.54$

The test uses a significance level ($\alpha$) of 0.05.

From the image, we have:

• Sample size ($n$): 367
• Number of successes (voters who prefer Candidate A): 178

Now, we need to calculate:

1. Test Statistic: We can use the formula for the test statistic in a proportion hypothesis test:

   $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

   where:

   • $\hat{p} = \frac{\text{number of successes}}{n} = \frac{178}{367}$
   • $p_0 = 0.54$ (the hypothesized population proportion)

2. p-value: Based on the calculated z-value, we can find the p-value for a left-tailed test.

3. Decision: Compare the p-value to $\alpha = 0.05$ to determine whether to reject the null hypothesis.

4. Conclusion: Depending on the test outcome, decide if there is sufficient evidence to support the claim that the proportion is less than 0.54.

I will perform these calculations for you now.from scipy.stats import norm

Given data

n = 367 # sample size successes = 178 # number of voters who prefer Candidate A p_0 = 0.54 # hypothesized proportion alpha = 0.05 # significance level

Sample proportion

p_hat = successes / n

Test statistic (z-value)

z = (p_hat - p_0) / ((p_0 * (1 - p_0) / n) ** 0.5)

p-value for a left-tailed test

p_value = norm.cdf(z)

z, p_value(-2.113549064066256, 0.017276900112459317)(-2.113549064066256, 0.017276900112459317)successHere are the results:

1. Test Statistic (z-value): $-2.11$
2. p-value: 0.0173 (rounded to four decimal places)

Now, let's interpret these values based on the significance level:

• The p-value (0.0173) is less than the significance level $\alpha = 0.05$.
• Therefore, we reject the null hypothesis.

Conclusion:

There is sufficient evidence to support the claim that the proportion of voters who prefer Candidate A is less than 0.54.

Would you like further details or have any questions?

Here are five follow-up questions for deeper understanding:

1. How does the choice of significance level affect hypothesis testing?
2. What would happen if the p-value were greater than $\alpha$?
3. Why is the z-test used for testing proportions in large samples?
4. How would the conclusion change if the sample size were different?
5. What does rejecting the null hypothesis imply about Candidate A's popularity?

Tip: When conducting hypothesis tests, always check if the conditions for using the test (like sample size) are met to ensure accurate results.